Reaction polyhedron

In the system (18) there exist laws of conservation (22) and it is imposed by the natural condition of having positive amounts (mole) of reactants. Hence it is possible to describe the region of composition spaces in which the solution for eqn. (18) N(t) (0 t oo) with non-negative initial conditions lies. It is a convex polyhedron specified by the set of linear equations and non-equalities [9, 10]. 

Let us designate this polyhedron as D(b) and call it a reaction polyhedron. Examples of the construction of D(b) for catalytic reactions are given below. 

Example 3. Let us consider a system of three isomers Ay, A2 and A3 taking part in catalytic isomerization reactions. 

Such a scheme for the catalytic isomerization of n-butenes over A1203 has been studied in detail previously [11]. Each reaction has a rate that is a function of both the gas composition and the surface state. In this case the assumption that the concentration of surface intermediates on the catalyst is a function of the gas composition is often used. It is a hypothesis about a quasi-steady state that is considered in detail in what follows. According to this hypothesis, for the reaction under study there exist three functions of the gas composition, w1w2, and w3, so that the kinetic equations can be written as 

It has been shown [11] that, even when we assume wt to be such linear 

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Butler, A. R., Glidewell, C., Hyde, A. R., and Walton, J. C. (1985). Formation of paramagnetic mononuclear iron nitrosyl complexes from diamagnetic di- and tetranuclear iron-sulphur nitrosyls Characterization by EPR spectroscopy and study of thiolate and nitrosyl ligand exchange reactions. Polyhedron 4, 797-809. 

The number of substances is five, therefore for every values of b1 and b2 the two equations (37) specify a three-dimensional plane (linear manifold in the space of compositions). Its intersection with the set of non-negative vectors (Nt 0, i = 1,.. 5) gives a three-dimensional reaction polyhedron. Let us... 

As a rule, in three-dimensional space the intersection of two straight lines is empty. In case it is not empty, then a small shift of one of the straight lines can make it empty. The intersection of a plane and a straight line is, as a rule, a point, and that of two planes is a straight line. To describe a reaction polyhedron, the first thing to do is to specify its nodes (vertices). They are the intersections of a plane specified by eqn. (37) with some faces of the set (a cone) of non-negative vectors. Faces of this cone are specified by the sets of equations and unequalities... 

To construct the reaction polyhedron D, it is insufficient to know only its vertices. We must also find its edges, i.e. one-parametric families of the positive solution for set (37) supplemented by the conditions Nt = 0 for a pair of indices t12. Two-dimensional (flat) faces are found as two-parametric families of solutions for eqn. (37) supplemented by the condition Nt = 0 with the only value of i. 

The equality to zero is obtained only in the case where, for any i,j = 1,.. ., n we have dJNfii = 8jlN, i.e. when the vector 3 (with components 8t) is proportional to that with components N t, in other words there exists a value of X such that 6, = XNf)t. But this is possible only in the case in which all the components 8t are simultaneously either positive or negative. Since, at some non-zero value of x, the vectors with components N0t and Noi + 3t must lie in the same reaction polyhedron, the simultaneous positivity or negativity for all the Si values is forbidden by, for example, the law of conservation of the overall (taking into account its adsorption) gas mass = Zm,-(iV0i + x3t) 1.171 1 = 0, for any Af we have m, > 0, hence <5 cannot have the same signs. Consequently, in the reaction polyhedron, G is a strictly convex function since the sum of a strictly convex G0 with a linear function of Df and a strictly convex function of IV5 is strictly convex in this polyhedron. 

The strict convexity of the function G in the reaction polyhedron D results in the following important property. In this polyhedron G has the unique local minimum. At the same time this local minimum is a global one. 

Note that the positive point for the G minimum in the reaction polyhedron is that of detailed equilibrium dG/dt < 0, hence at the point of minimum we have dG/d< = 0 (a decrease is possible "no-where ) and according to eqns. (83)-(85), dG/dt = 0 only at the PDEs. 

These notes suggest that each reaction polyhedron has a positive PDE coinciding with a point at which G is a minimum (we assumed the existence of a positive point at least in one reaction polyhedron and as a consequence the existence of such a point for every polyhedron). 

It is clearly seen that, at a vertex of the reaction polyhedron, G achieves its local maximum value (due to the strict convexity of G and the fact that its minimum point is positive). Therefore near each vertex, as well as in the vicinity of some faces, the G function can be used to construct a region that is unattainable from outside. Let us consider the case of one vertex and then a more awkward general situation. 

Let be a vertex for the reaction polyhedron with outcoming edges d1,... 

We will now describe the construction of an "unattainability region near the arbitrary multitude of vertices. Let it be a multitude E for the vertices of the reaction polyhedron. P(E) will be a multitude of D edges connecting vertices from E, and K(E) are those connecting elements E with vertices not belonging to E. As before, let Md = min G(N) be a minimum G... 

An operation is constructed that associates each closed multitude M from a reaction polyhedron with the other one J0(M)... 

In each reaction polyhedron, the region specified by the inequality... 

These regions in all reaction polyhedra can be described by the same inequality. For this purpose let us recall (Sect. 2) that we constructed G(c) using an arbitrary PDE not necessarily lying in the examined reaction polyhedron and showed that this function is a Lyapunov function for any reaction polyhedron. Now let us introduce one more Lyapunov function which differs from the previous one in every reaction polyhedron by a constant, depending, nevertheless, on this polyhedron. Let us prescribe a function c (c) whose value is PDE accounting for the initial conditions c (lying in the same reaction polyhedron). Let us determine... 

We know that a PDE is stable as a linear approximation (see Sect. 2). Whence from eqns. (137) and (138) we establish that, at sufficiently low um and vout and t - oo, a solution of the kinetic equations for homogeneous systems tends to a unique steady-state point localized inside the reaction polyhedron with balance relationships (138) in a small vicinity of a positive PDE. If b(c(0)) = 6(c,n) vinjvout, then at low v,n and eout the function c(t) is close to the time dependence of concentrations for a corresponding closed system. To be more precise, if vm -> 0, uout -> 0, vmjvOM, c(0), cin are constant and c(O) is not a boundary PDE, then we obtain max c(t) — cc](t) -> 0, where ccl(t) is the solution of the kinetic equations for closed systems, ccl(0) = c(0),and is the Euclidian norm in the concentration space. 

Studies of linear systems and systems without "intermediate interactions show that a positive steady state is unique and stable not only in the "thermodynamic case (closed systems). Horn and Jackson [50] suggested one more class of chemical kinetic equations possessing "quasi-ther-modynamic properties, implying that a positive steady state is unique and stable in a reaction polyhedron and there exist a global (throughout a given polyhedron) Lyapunov function. This class contains equations for closed systems, linear mechanisms, and intersects with a class of equations for "no intermediate interactions reactions, but does not exhaust it. Let us describe the Horn and Jackson approach. 

Studies [96, 98, 102] of the number of steady-state points in eqns. (5) belonging to the reaction polyhedron (simplex) 3... 

Localization of this steady state as a point of intercept for the null dines x = 0 and y = 0 as a function of the k x value is shown in Fig. 16. At low k x this point is localized sufficiently close to the region of probable initial conditions (at k x = 0 it becomes a boundary steady state). It is the proximity of the initial conditions to the steady state outside the reaction polyhedron that accounts for the slow transition regime. Note that, besides two real-valued steady states, the system also has two complex-valued steady states. At bifurcation values of the parameters, the latter become real and appear in the reaction simplex as an unrough internal steady state. The proximity of complex-valued roots of the system to the reaction simplex also accounts for the generation of slow relaxations. 

Thus the effect of slow relaxations can also be noticeable in the case when the steady state inside the reaction polyhedron is unique and stable as a whole (all positive solutions tend to it at t - oo). For this purpose it suffices that the "external (non-physical) steady state is close to the polyhedron boundary and the initial conditions localize on the opposite side of the boundary (inside the polyhedron). 

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Wang, H.X., Wu, H.R, Yang, X.L. et al. (2007) Highly active ferrocenylamine-derived pallada-cycles for carbon-carbon cross-coupling reactions. Polyhedron, 26, 3857-64. 

Darensbourg DJ, Chung WC (2013) Relative basicities of cyclic ethers and esters. Chemistry of importance to ring-operting co- and teipolymerization reactions. Polyhedron 58 139-143... 

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